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On the Boltzmann equation for charged particle beams under the effect of strong magnetic fields
1. | Institut de Mathématiques de Marseille UMR 7353, Aix Marseille Université, CNRS, Centrale Marseille, 13453 Marseille, France |
References:
[1] |
M. Bostan, The Vlasov-Poisson system with strong external magnetic field. Finite Larmor radius regime, Asymptot. Anal., 61 (2009), 91-123. |
[2] |
M. Bostan, Transport equations with disparate advection fields. Application to the gyrokinetic models in plasma physics, J. Differential Equations, 249 (2010), 1620-1663.
doi: 10.1016/j.jde.2010.07.010. |
[3] |
M. Bostan, Gyrokinetic Vlasov equation in three dimensional setting. Second order approximation, SIAM J. Multiscale Model. Simul., 8 (2010), 1923-1957.
doi: 10.1137/090777621. |
[4] |
M. Bostan and C. Caldini-Queiros, Finite Larmor radius approximation for collisional magnetized plasmas, C. R. Math. Acad. Sci. Paris, 350 (2012), 879-884.
doi: 10.1016/j.crma.2012.09.019. |
[5] |
M. Bostan and C. Caldini-Queiros, Finite Larmor radius approximation for collisional magnetic confinement. Part I: The linear Boltzmann equation, Quart. Appl. Math., 72 (2014), 323-345.
doi: 10.1090/S0033-569X-2014-01356-1. |
[6] |
M. Bostan and C. Caldini-Queiros, Finite Larmor radius approximation for collisional magnetic confinement. Part II: The Fokker-Planck-Landau equation, to appear in Quart. Appl. Math. |
[7] |
A. J. Brizard, A guiding-center Fokker-Planck collision operator for nonuniform magnetic fields, Phys. Plasmas, 11 (2004), 4429-4438.
doi: 10.1063/1.1780532. |
[8] |
A. J. Brizard and T. S. Hahm, Foundations of nonlinear gyrokinetic theory, Rev. Modern Phys., 79 (2007), 421-468.
doi: 10.1103/RevModPhys.79.421. |
[9] |
C. Cercignani, The Boltzmann Equation and Its Applications, Springer-Verlag New-York 1988.
doi: 10.1007/978-1-4612-1039-9. |
[10] |
C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Applied Mathematical Sciences, 106, Springer-Verlag New-York, 1994.
doi: 10.1007/978-1-4419-8524-8. |
[11] |
R. C. Davidson and H. Qin, Physics of Charged Particle Beams in High Energy Accelerators, Imperial College Press, World Scientific Singapore, 2001.
doi: 10.1142/p250. |
[12] |
P. Degond and P.-A. Raviart, On the paraxial approximation of the stationary Vlasov-Maxwell system, Math. Models Meth. Appl. Sci., 3 (1993), 513-562.
doi: 10.1142/S0218202593000278. |
[13] |
F. Filbet and E. Sonnendrücker, Modeling and numerical simulation of space charge dominated beams in the paraxial approximation, Math. Models Methods Appl. Sci., 16 (2006), 763-791.
doi: 10.1142/S0218202506001340. |
[14] |
E. Frénod, Application of the averaging method to the gyrokinetic plasma, Asymptot. Anal., 46 (2006), 1-28. |
[15] |
E. Frénod and A. Mouton, Two-dimensional finite Larmor radius approximation in canonical gyrokinetic coordinates, J. Pures Appl. Math. Adv. Appl., 4 (2010), 135-169. |
[16] |
E. Frénod and E. Sonnendrücker, Homogenization of the Vlasov equation and of the Vlasov-Poisson system with strong external magnetic field, Asymptotic Anal., 18 (1998), 193-213. |
[17] |
E. Frénod and E. Sonnendrücker, The finite Larmor radius approximation, SIAM J. Math. Anal., 32 (2001), 1227-1247.
doi: 10.1137/S0036141099364243. |
[18] |
X. Garbet, G. Dif-Pradalier, C. Nguyen, Y. Sarazin, V. Grandgirard and Ph. Ghendrih, Neoclassical equilibrium in gyrokinetic simulations, Phys, Plasmas, 16 (2009), 062503.
doi: 10.1063/1.3153328. |
[19] |
F. Golse and L. Saint-Raymond, The Vlasov-Poisson system with strong magnetic field, J. Math. Pures Appl., 78 (1999), 791-817.
doi: 10.1016/S0021-7824(99)00021-5. |
[20] |
G. Laval, S. Mas-Gallic and P.-A. Raviart, Paraxial approximation of ultra-relativistic intense beams, Numer. Math., 69 (1994), 33-60.
doi: 10.1007/s002110050079. |
[21] |
D. Levermore, Moment closure hierarchies for kinetic theories, J. Statist. Phys., 83 (1996), 1021-1065.
doi: 10.1007/BF02179552. |
[22] |
J. Madsen, Gyrokinetic linearized Landau collision operator, Phys. Review, 87 (2013), 011101.
doi: 10.1103/PhysRevE.87.011101. |
[23] |
P.-A. Raviart, Paraxial approximation of the stationary Vlasov-Maxwell equations, in Nonlinear Partial Differential Equations and Their Applications, Collège de France Seminar, Vol. XIII, Paris, (1991-1993), Pitman Res. Notes Math. Ser., 302, Longman Sci. Tech., Harlow, 1994, 158-171. |
[24] |
H. Tanaka, Probabilistic treatment of the Boltzmann equation of Maxwellian molecules, Z. Wahrsch. Verw. Gebiele, 46 (1978/79), 67-105.
doi: 10.1007/BF00535689. |
[25] |
G. Toscani and C. Villani, Probability metrics and uniqueness of the solution to the Boltzmann equation for a Maxwell gas, J. Statist. Phys., 94 (1999), 619-637.
doi: 10.1023/A:1004589506756. |
[26] |
G. Toscani and C. Villani, Sharp entropy dissipation bounds and explicit rate of trend to equilibrium for the spatially homogeneous Boltzmann equation, Comm. Math. Phys., 203 (1999), 667-706.
doi: 10.1007/s002200050631. |
[27] |
C. Villani, On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations, Arch. Rational Mech. Anal., 143 (1998), 273-307.
doi: 10.1007/s002050050106. |
[28] |
C. Villani, Contribution à l'étude mathématique des collisions en théorie cinétique, Master's thesis, Université Paris-Dauphine France, 2000. |
[29] |
X. Q. Xu and M. N. Rosenbluth, Numerical simulation of ion-temperature-gradient-driven modes, Phys. Fluids B, 3 (1991), 627-643.
doi: 10.1063/1.859862. |
show all references
References:
[1] |
M. Bostan, The Vlasov-Poisson system with strong external magnetic field. Finite Larmor radius regime, Asymptot. Anal., 61 (2009), 91-123. |
[2] |
M. Bostan, Transport equations with disparate advection fields. Application to the gyrokinetic models in plasma physics, J. Differential Equations, 249 (2010), 1620-1663.
doi: 10.1016/j.jde.2010.07.010. |
[3] |
M. Bostan, Gyrokinetic Vlasov equation in three dimensional setting. Second order approximation, SIAM J. Multiscale Model. Simul., 8 (2010), 1923-1957.
doi: 10.1137/090777621. |
[4] |
M. Bostan and C. Caldini-Queiros, Finite Larmor radius approximation for collisional magnetized plasmas, C. R. Math. Acad. Sci. Paris, 350 (2012), 879-884.
doi: 10.1016/j.crma.2012.09.019. |
[5] |
M. Bostan and C. Caldini-Queiros, Finite Larmor radius approximation for collisional magnetic confinement. Part I: The linear Boltzmann equation, Quart. Appl. Math., 72 (2014), 323-345.
doi: 10.1090/S0033-569X-2014-01356-1. |
[6] |
M. Bostan and C. Caldini-Queiros, Finite Larmor radius approximation for collisional magnetic confinement. Part II: The Fokker-Planck-Landau equation, to appear in Quart. Appl. Math. |
[7] |
A. J. Brizard, A guiding-center Fokker-Planck collision operator for nonuniform magnetic fields, Phys. Plasmas, 11 (2004), 4429-4438.
doi: 10.1063/1.1780532. |
[8] |
A. J. Brizard and T. S. Hahm, Foundations of nonlinear gyrokinetic theory, Rev. Modern Phys., 79 (2007), 421-468.
doi: 10.1103/RevModPhys.79.421. |
[9] |
C. Cercignani, The Boltzmann Equation and Its Applications, Springer-Verlag New-York 1988.
doi: 10.1007/978-1-4612-1039-9. |
[10] |
C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Applied Mathematical Sciences, 106, Springer-Verlag New-York, 1994.
doi: 10.1007/978-1-4419-8524-8. |
[11] |
R. C. Davidson and H. Qin, Physics of Charged Particle Beams in High Energy Accelerators, Imperial College Press, World Scientific Singapore, 2001.
doi: 10.1142/p250. |
[12] |
P. Degond and P.-A. Raviart, On the paraxial approximation of the stationary Vlasov-Maxwell system, Math. Models Meth. Appl. Sci., 3 (1993), 513-562.
doi: 10.1142/S0218202593000278. |
[13] |
F. Filbet and E. Sonnendrücker, Modeling and numerical simulation of space charge dominated beams in the paraxial approximation, Math. Models Methods Appl. Sci., 16 (2006), 763-791.
doi: 10.1142/S0218202506001340. |
[14] |
E. Frénod, Application of the averaging method to the gyrokinetic plasma, Asymptot. Anal., 46 (2006), 1-28. |
[15] |
E. Frénod and A. Mouton, Two-dimensional finite Larmor radius approximation in canonical gyrokinetic coordinates, J. Pures Appl. Math. Adv. Appl., 4 (2010), 135-169. |
[16] |
E. Frénod and E. Sonnendrücker, Homogenization of the Vlasov equation and of the Vlasov-Poisson system with strong external magnetic field, Asymptotic Anal., 18 (1998), 193-213. |
[17] |
E. Frénod and E. Sonnendrücker, The finite Larmor radius approximation, SIAM J. Math. Anal., 32 (2001), 1227-1247.
doi: 10.1137/S0036141099364243. |
[18] |
X. Garbet, G. Dif-Pradalier, C. Nguyen, Y. Sarazin, V. Grandgirard and Ph. Ghendrih, Neoclassical equilibrium in gyrokinetic simulations, Phys, Plasmas, 16 (2009), 062503.
doi: 10.1063/1.3153328. |
[19] |
F. Golse and L. Saint-Raymond, The Vlasov-Poisson system with strong magnetic field, J. Math. Pures Appl., 78 (1999), 791-817.
doi: 10.1016/S0021-7824(99)00021-5. |
[20] |
G. Laval, S. Mas-Gallic and P.-A. Raviart, Paraxial approximation of ultra-relativistic intense beams, Numer. Math., 69 (1994), 33-60.
doi: 10.1007/s002110050079. |
[21] |
D. Levermore, Moment closure hierarchies for kinetic theories, J. Statist. Phys., 83 (1996), 1021-1065.
doi: 10.1007/BF02179552. |
[22] |
J. Madsen, Gyrokinetic linearized Landau collision operator, Phys. Review, 87 (2013), 011101.
doi: 10.1103/PhysRevE.87.011101. |
[23] |
P.-A. Raviart, Paraxial approximation of the stationary Vlasov-Maxwell equations, in Nonlinear Partial Differential Equations and Their Applications, Collège de France Seminar, Vol. XIII, Paris, (1991-1993), Pitman Res. Notes Math. Ser., 302, Longman Sci. Tech., Harlow, 1994, 158-171. |
[24] |
H. Tanaka, Probabilistic treatment of the Boltzmann equation of Maxwellian molecules, Z. Wahrsch. Verw. Gebiele, 46 (1978/79), 67-105.
doi: 10.1007/BF00535689. |
[25] |
G. Toscani and C. Villani, Probability metrics and uniqueness of the solution to the Boltzmann equation for a Maxwell gas, J. Statist. Phys., 94 (1999), 619-637.
doi: 10.1023/A:1004589506756. |
[26] |
G. Toscani and C. Villani, Sharp entropy dissipation bounds and explicit rate of trend to equilibrium for the spatially homogeneous Boltzmann equation, Comm. Math. Phys., 203 (1999), 667-706.
doi: 10.1007/s002200050631. |
[27] |
C. Villani, On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations, Arch. Rational Mech. Anal., 143 (1998), 273-307.
doi: 10.1007/s002050050106. |
[28] |
C. Villani, Contribution à l'étude mathématique des collisions en théorie cinétique, Master's thesis, Université Paris-Dauphine France, 2000. |
[29] |
X. Q. Xu and M. N. Rosenbluth, Numerical simulation of ion-temperature-gradient-driven modes, Phys. Fluids B, 3 (1991), 627-643.
doi: 10.1063/1.859862. |
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